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Human mathematicians are being outcounterexampled

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Re: Human mathematicians are being outcounterexampled

#31

Post by simonreiff »

tchalla wrote: post #24
hintymad wrote: post #14

> The Jacobian Conjecture Interestingly, Yitang Zhang of the twin-prime-conjecture fame spent 7 years working on the Jacobian conjecture under the advisor Tzuong-Tsieng Moh at Purdue. A key step in his thesis used a corollary of Moh's. It turned out that the corollary was incorrect. As a result, Moh refused to write any recommendation letter for Zhang, and Zhang couldn't find any teaching or research job and ended up…

I was once in a presentation for a math PhD thesis. During the thesis, the evaluator of the thesis noticed a flaw in their proof. The student understood and then asked “What now?” The evaluator prof simply shrugged.

That sounds like the worst "exam nightmare" scenario imaginable, but did the student get the PhD in the end?

Re: Human mathematicians are being outcounterexampled

#32

Post by SpicyLemonZest »

mb7733 wrote: post #25
parpfish wrote: post #11

Earlier quoted context omitted.

proofs by counterexample are effective but ultimately unsatisfying. they get you to an answer but they don't help help you understand and bend you r mind into seeing how the math works and lead you on to the new set of questions. and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians

Are you thinking of proof by contradiction, which is rejected by constructionism? [Dis]proof by counterexample is the most straightforward way to show a statement to be false. What better way is there to disprove a general statement like 'all x are y' than finding an 'x' that isn't 'y'?

It’s very straightforward, but it often doesn’t (and here didn’t) fully satisfy the curiosity that was embedded in the original problem. Why did the Jacobian conjecture seem to be true? Is there some underlying symmetry that’s very slightly broken? Or perhaps there’s all kinds of counterexamples, and the intuitive pattern is only real for certain kinds of functions which happen to predominate in our intuition. Then how should we adjust our intuitions to better capture the space of possible polynomial functions?

Re: Human mathematicians are being outcounterexampled

#33

Post by soupspaces »

satvikpendem wrote: post #5

That's a good thing. It saves people wasting time trying to prove something they now know to be false, so that they can move on to other things to prove, it's a more fruitful use of humanity's time overall at least in the field of mathematics.

Except you can't possibly know that. New insight can arise regardless of whether mathematicians are trying to prove or disprove a statement, and regardless of whether the statement ultimately turns out to be true or false.

Re: Human mathematicians are being outcounterexampled

#34

Post by bananaflag »

taneq wrote: post #22
parpfish wrote: post #11

Earlier quoted context omitted.

proofs by counterexample are effective but ultimately unsatisfying. they get you to an answer but they don't help help you understand and bend you r mind into seeing how the math works and lead you on to the new set of questions. and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians

Maybe I’m just not pure enough but I find the whole concept of proof by counterexample to be elegant, and I don’t see why proving that something must be true is superior to proving that it can’t be false.

You mean proof by contradiction, which is something different.

Re: Human mathematicians are being outcounterexampled

#35

Post by OG_BME »

QuesnayJr wrote: post #6

If the poster's (is it Kevin Buzzard?) suggestion works out and AI finds a counterexample to the Hodge conjecture, that would be a really big deal. It's one of the Millenium problems, for example. One thing that he mentions that already quite surprising is that AI was able to autoformalize the Golod-Shaferevich theorem and proof.

I think he was being provocative and maybe a bit tongue-in-cheek when he said that. A candidate object alone doesn't resolve the Hodge Conjecture. Any apparent counterexample would have to prove that no algebraic cycle exists, no invariant subspace exists, or that every element of an infinite ideal is nilpotent. Much harder, but not impossible.

Re: Human mathematicians are being outcounterexampled

#36

Post by hintymad »

derbOac wrote: post #23
hintymad wrote: post #14

> The Jacobian Conjecture Interestingly, Yitang Zhang of the twin-prime-conjecture fame spent 7 years working on the Jacobian conjecture under the advisor Tzuong-Tsieng Moh at Purdue. A key step in his thesis used a corollary of Moh's. It turned out that the corollary was incorrect. As a result, Moh refused to write any recommendation letter for Zhang, and Zhang couldn't find any teaching or research job and ended up…

Inspiring? Because of the twin prime conjecture success following his time in the wilderness? I suppose so. I'm tired of tales like this in academics though. That's not a criticism of you for telling the tale, I'm just so tired of this kind of thing in academics in general. So, so, so much politics and public reputation management. Zhang should have never had to suffer like that. As my own research has drifted more i…

> So, so, so much politics and public reputation management. Zhang should have never had to suffer like that.

Very true. Unfortunately, when there are people, there will be politics. I remember when reading Yau's autobiography, I kept marvel how much calculation, or "politics" if you will, that Yau mentioned or implied in the book.

> My guess is the outcome would have been the same for the same reasons?

At least Zhang didn't have to spend 7 years working on the Jacobian conjecture. He said in an interview that he always wanted to work on number theory. Moh asked him to work on Jacobian, and he obliged.

Re: Human mathematicians are being outcounterexampled

#38

Post by bananaflag »

Dove wrote: post #15

When I was in grad school, I had the opportunity to take a course from my adviser in which he discussed his current research and some open questions. It was a relatively accessible subject area and the questions were sometimes easy enough that we could meaningfully contribute. On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. It…

> It was also the case that I had absolutely no command of the sort of machinery that one would use to prove such a thing, but I could certainly look for a counterexample.

Hm, as a mathematician, my experience feels opposite. A proof would be an adaptation of a proof I know, some tweaking it here and there. A counterexample would require some deep understanding of the structure of the objects involved, which frequently is beyond my comprehension.

But probably this is because I think of quite abstract objects which are harder to grasp. For numbers or polynomials, this would be the other way round.

Re: Human mathematicians are being outcounterexampled

#39

Post by remus »

parpfish wrote: post #11
satvikpendem wrote: post #5

That's a good thing. It saves people wasting time trying to prove something they now know to be false, so that they can move on to other things to prove, it's a more fruitful use of humanity's time overall at least in the field of mathematics.

proofs by counterexample are effective but ultimately unsatisfying. they get you to an answer but they don't help help you understand and bend you r mind into seeing how the math works and lead you on to the new set of questions. and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians

I would only agree partially. There are counterexamples that are not illustrative, but it is fairly common that in thinking about how to construct a counterexample you gain a more thorough understanding of the original problem and at least one fundamental issue which prevents the conjecture from being true.

Re: Human mathematicians are being outcounterexampled

#40

Post by dzdt »

I suppose it will fall to AI as well to compose the mathematical equivalent of The Ballad of John Henry. Who will be the human champion, the last great hero who can deliver proofs "from the book" that a machine cannot outperform?

[1] https://en.wikipedia.org/wiki/John_Henry_(folklore)

[2] https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK

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